Single crystal micromechanical resonator and fabrication methods thereof

ABSTRACT

The present invention relates to a single crystal micromechanical resonator. In particular, the resonator includes a lithium niobate or lithium tantalate suspended plate. Also provided are improved microfabrication methods of making resonators, which does not rely on complicated wafer bonding, layer fracturing, and mechanical polishing steps. Rather, the methods allow the resonator and its components to be formed from a single crystal.

STATEMENT OF GOVERNMENT INTEREST

This invention was made with Government support under contract no.DE-AC04-94AL85000 awarded by the U.S. Department of Energy to SandiaCorporation. The Government has certain rights in the invention.

FIELD OF THE INVENTION

The present invention relates to a single crystal micromechanicalresonator and microfabrication methods thereof. In particular, theresonator is formed by ion implantation of a lithium niobate or lithiumtantalate single crystal.

BACKGROUND OF THE INVENTION

Micromechanical resonators (or microresonators) are miniature acousticresonators fabricated using integrated circuit microfabricationtechniques. Such microresonators have various uses, such as filters andoscillators. In particular, the resonant frequency can be definedphotolithographically, thereby allowing numerous filters spanning fromseveral hundred MHz to several GHz to be realized on a single chip.

Currently, band select filters in cellular handsets are realized using acombination of many dies containing bulk (BAW) or surface (SAW) acousticwave resonators. The resonant frequency for these BAW resonators isdetermined generally by the thickness of a deposited thin film andrequires a separate film thickness for each filter frequency. This makesintegration of multiple frequency filters on a single die bothchallenging and costly. While in theory SAW resonators can support awide range of frequencies on a single chip, in practice, the thicknessof the metal interdigitated electrodes used to transduce a SAW resonatoris varied with frequency, thereby limiting the range of filter bandsthat can be covered on a single chip.

By basing the resonance on a laterally propagating Lamb wave in asuspended plate with a thickness less than an acoustic wavelength, awide range of filter frequencies can be achieved on a single wafer byaltering the CAD-layout of the devices. Piezoelectric Lamb waveresonators formed in deposited thin films of aluminum nitride, zincoxide, and lead zirconate titanate, while having much higher couplingcoefficients than electrostatically transduced microresonators, still donot have a high enough coupling coefficient for many of the band selectfilters in wireless handsets. Thus, there is a need for improvedmicroresonators having sufficiently high coupling coefficients andCAD-definable frequencies, as well as new methods for fabricating suchmicroresonators on a single chip.

SUMMARY OF THE INVENTION

Accordingly, the present invention relates to a Lamb wave microresonatorhaving a resonant frequency defined by the plate width W, which is aphotolithographically definable dimension of the device. The presentinvention also provides other lithographically definable resonator andelectrode dimensions (e.g., any dimension described herein, such asresonator plate thickness t or electrode aperture a) that can beoptimized to provide desired device characteristics.

In particular embodiments, the microresonator of the invention hasnumerous improved characteristics, such as a high coupling coefficientk_(eff) ², a high quality factor Q, and a high resonator figure of meritFOM. Such characteristics can be improved, e.g., by matching the crystalorientation with the type of plate acoustic wave (e.g., SH0 or S0)having the highest electromechanical coupling coefficient K² or k_(eff)² for that orientation, by controlling the electrode spacing or aperturea, by arraying numerous resonators in parallel, as well as any otherdesign considerations described herein.

Also described herein are methods of fabricating such microresonators ina single crystal in crystallographic class 3 m (e.g., lithium niobate orlithium tantalate). The method includes forming a damaged layer beneaththe top surface of the crystal, providing at least one trench to accessthat damaged layer, and then removing the damaged layer with an etchant.In this manner, the resonating portion, the support structure, and theanchoring regions configured to suspend the resonating portion areformed within the same single crystal. In particular, the methods of theinvention do not require costly wafer bonding, polishing, or fracturingprocesses.

In a first aspect, the invention features a method for fabricating amicromechanical resonator, the method including: (i) providing a single3 m crystal (e.g., lithium niobate or lithium tantalate); (ii) treatingan exposed area of the single crystal with ions, thereby creating an iondamaged region below the top surface of the crystal; (iii) providing atleast one trench that defines a first dimension of the resonator; and(iv) removing the ion damaged layer with an etchant. In someembodiments, the method thereby releases at least a resonating portionof the resonator from the crystal.

In some embodiments, the method includes, before step (ii), patterning atop surface of the crystal with a mask, thereby defining the exposedarea. In other embodiments, the mask includes a plurality of exposedareas. In further embodiments, each exposed area defines a resonatingportion of a resonator, thereby providing a plurality of micromechanicalresonators on a single die. In yet other embodiments, two or more of theplurality of micromechanical resonators are the same or different.

In some embodiments, the method includes, before step (iv), depositing aprotective layer on the first dimension of the resonator. In furtherembodiments, the method includes stripping the protective layer (e.g.,after step (iv)).

In other embodiments, the method includes, after step (i), depositing ametal layer on a top surface of the crystal, where the mask is thenpatterned on top of the metal layer. In further embodiments, the methodincludes patterning the metal layer with one or more electrodes.

In some embodiments, the method includes, after step (iv), annealing theresonator (e.g., thereby healing the ion damaged layer).

In some embodiments, the crystal is an X-cut lithium niobate crystal, aY-cut lithium niobate crystal, a Z-cut lithium niobate crystal, arotated cut lithium niobate crystal, an X-cut lithium tantalate crystal,a Y-cut lithium tantalate crystal, a Z-cut lithium tantalate crystal, ora rotated-cut lithium tantalate crystal (e.g., a rotated X-, Y-, orZ-cut crystal, where the rotation can be of any useful angle, such as136°).

In other embodiments, the resonator is a shear mode Lamb wave resonatoror a symmetric mode Lamb wave resonator.

In some embodiments, the ion is helium or hydrogen.

In a second aspect, the invention features a single crystalmicromechanical resonator including a resonating portion that includes asingle X-cut or Y-cut crystal of lithium niobate or lithium tantalate; asupport structure disposed below the resonating portion, where theresonating portion and the support structure are composed of the samesingle X-cut or Y-cut crystal; and a plurality of anchoring regionsconfigured to suspend the resonating portion within the supportstructure, where a trench is disposed below and/or around the resonatingportion.

In some embodiments, the plurality of anchoring regions, resonatingportion, and the support structure are composed of the same single X-cutor Y-cut crystal.

In other embodiments, the resonator further includes a plurality ofelectrodes disposed on a surface of the resonating portion. In yet otherembodiments, the resonating portion includes a suspended plate.

In a third aspect, the invention features a die including a plurality ofsingle crystal micromechanical resonators (e.g., any described herein).In some embodiments, two or more resonators are the same or different.

In a fourth aspect, the invention features a band select filter arrayincluding a plurality of single crystal micromechanical resonators(e.g., any described herein). In some embodiments, two or moreresonators are the same or different.

Definitions

As used herein, the term “about” means +/−10% of any recited value. Asused herein, this term modifies any recited value, range of values, orendpoints of one or more ranges.

By “micro” is meant having at least one dimension that is less than 1mm. For instance, a microresonator can have a length, width, height,cross-sectional dimension, circumference, radius (e.g., external orinternal radius), or diameter that is less than 1 mm.

As used herein, the terms “top,” “bottom,” “upper,” “lower,” “above,”and “below” are used to provide a relative relationship betweenstructures. The use of these terms does not indicate or require that aparticular structure must be located at a particular location in theapparatus.

Other features and advantages of the invention will be apparent from thefollowing description and the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A-1C shows schematics of an exemplary microresonator. Provided are(A) an exemplary microresonator 100 having a plate suspended over atrench 150, (B) a cross section view along section line 1-1, and (C) aperspective view showing various dimensions of the microresonator,including width W, length L, and thickness t of the resonating portion,as well as electrode gap g, electrode aperture a, electrode width e, andspace between the electrode and the edge of the resonating portion s.

FIG. 2 is a top down schematic of an X-cut LiNbO₃ wafer showing the z-and y-crystal planes. Gray rectangles represent the microresonatorhaving two electrodes (parallel black lines). Shown are angles ofacoustic propagation relative to the z- or y-axis. The couplingcoefficient and sound velocity for each Lamb mode vary with thedirection of acoustic propagation.

FIG. 3A-3C shows exemplary process flows used to fabricate amicroresonator 200, 300, 400. Provided are (A) a schematic of anexemplary process for a microresonator 200; (B) a schematic of anexemplary process for a microresonator having electrodes 300; and (C)top views (left) and cross section views (right) of the microresonator400 after steps (a)-(f), as described herein.

FIG. 4A-4B shows scanning electron microscope images of a LiNbO₃microresonator rotated 80° to the z-axis (or 170° to the y-axis). Thedesigned microresonator dimensions are for a width W=20 μm, length L=140μm, and an aperture (or electrode overlap) a=50 μm. Each electrode is 5μm wide, and the gap g between electrodes is also 5 μm. Provided is (A)a top view and (B) a close-up view of a released LiNbO₃ microresonator.

FIG. 5A-5B shows optical images of a LiNbO₃ microresonator rotated atdifferent angles. Provided are LiNbO₃ microresonators (A) rotated 80° tothe z-axis (170° to the y-axis) and (B) rotated 70° to the z-axis (160°to the y-axis) just prior to release. In FIG. 5B, the optional Au layerwas used to protect the −z face of the LiNbO₃ microresonator sidewall.

FIG. 6 is a cross section image after chlorine dry etching 1.32 μm ofX-cut LiNbO₃ rotated 80° from the z-axis (170° from the y-axis) using anoxide hard mask. The sidewall angle is in excess of 84°.

FIG. 7 is a finite element modeling (FEM) showing the displacementprofile of the mode shape, on resonance, for the SH0 Lamb wave in thedevice shown in FIG. 4A.

FIG. 8 shows simulated admittance versus frequency using finite elementmodeling of the SH0 mode LiNbO₃ microresonator with a quality factor of100, 5 μm wide electrodes, and a varying electrode gap. Provided aresimulations with a 5 μm layer of vacuum surrounding the resonator. Thecoupling coefficient is seen to increase for larger electrode gapsindicating optimum placement of the electrodes at the anti-nodes.

FIG. 9 shows simulated coupling coefficient versus electrode gap usingfinite element modeling of the SH0 mode LiNbO₃ microresonator with aquality factor of 100 and varying electrode widths. The couplingcoefficient is seen to increase for larger electrode gaps indicatingoptimum placement of the electrodes at the anti-nodes and to be weaklydependent on the electrode width. The solid lines are for finite elementsimulations with a 5 μm thick layer of vacuum above and below theresonator, while the dashed lines are for simulations omitting thedielectric properties above and below the device. Omitting vacuum aroundthe resonator overestimates the coupling coefficient by as much as 1.8%.

FIG. 10 shows measured narrow band admittance (black line ii) of the SH0lamb wave LiNbO₃ microresonator rotated 80° from the z-axis (170° fromthe y-axis) shown in FIG. 4A. Line (i) shows the admittance of theelectrical equivalent circuit model in FIG. 12 with a frequency of100.965 Mhz, a motional impedance of 1076Ω, a quality factor of 1300,and a k_(t) ² of 17.5%. Line (iii) shows the predicted resonatoradmittance from finite element modeling.

FIG. 11 shows measured wide band admittance (black line ii) of the SH0lamb wave LiNbO₃ microresonator rotated 80° from the z-axis (170° fromthe y-axis) shown in FIG. 4A. Line (i) shows the admittance of theelectrical equivalent circuit model in FIG. 12 with a frequency of100.965 Mhz, a motional impedance of 1076Ω, a quality factor of 1300,and a k_(t) ² of 17.5%. Line (iii) shows the measured response betweentwo bond pads rotated 170° from the y-axis, verifying that transductionfrom the bond pads is the source of many of the spurious modes seen inthe resonator response.

FIG. 12 shows a Butterworth-Van Dyke equivalent circuit model of anacoustic resonator.

FIG. 13 shows measured resonant frequency versus temperature for theLiNbO₃ microresonator shown in FIG. 4A.

FIG. 14 shows measured admittance of LiNbO₃ SH0 Lamb wave resonatorswith acoustic propagation rotated (i) 60°, (ii) 70°, (iii) 80°, and(iii) 90° from the z-axis (i.e., rotated (i) 150°, (ii) 160°, (iii)170°, and (iii) 180° from the y-axis, respectively). The higher 4; ofthe 80° rotated sample is clearly visible from the measured admittances.

FIG. 15 is a FEM showing displacement (Full) and strain (Half) profilesof the S0 and SH0 resonator modes for a microresonator having anaperture a=50 μm.

FIG. 16 is a graph showing FEM and experimental 4; versus aperture.

FIG. 17 shows a modified Butterworth-Van Dyke equivalent circuit modelof an acoustic resonator.

FIG. 18 shows FEM and experimental results for the S0 Lamb waveresonator with an aperture a=90 μm.

FIG. 19 shows FEM and experimental results for the SH0 Lamb waveresonator with an aperture a=90 μm.

FIG. 20A-20B shows experimental (black) and equivalent circuit model(gray) admittance for a SH0 Lamb wave resonator with a plate width W=4.4μm.

DETAILED DESCRIPTION OF THE INVENTION

The present invention relates to a micromechanical resonator (ormicroresonator) having a high coupling coefficient k_(eff) ². Inparticular, the resonator includes a resonating portion, one or moreelectrodes disposed on at least one surface of the resonating portion,and a support structure disposed below the resonating portion. Infurther embodiments, the resonating portion includes one or moreanchoring regions, which suspend the resonating portion within thesupport structure. In yet other embodiments, the suspended resonatingportion is surrounded by a trench, which separates the resonatingportion and the support structure. Additional details of all thesecomponents are described herein.

FIG. 1A-1B shows an exemplary microresonator. As seen in FIG. 1A, themicroresonator 100 includes a resonating portion 111 and two anchoringregion 112, 113 configured to suspend the resonating portion 111 withinthe support structure 110. A pair of electrodes 121, 122 on the topsurface of the resonating portion 111 is electrically connected tocontact pads 141, 142 by wiring 131, 132. Ground electrodes 161, 162 arealso disposed on the top face of the microresonator 100. The electrodescan have any useful geometry (e.g., bars, interdigitated fingers, etc.)and useful location (e.g., on the top and/or bottom surfaces of themicroresonator, resonating portion, and/or anchoring region(s)).

The resonating portion can be suspended. For example, the microresonator100 includes a trench 150, which is located between the bottom surfaceof the resonating portion 111 and a surface of the support structure110. As seen in FIG. 1B, the trench 150 is disposed below and around theresonating portion 111. In this manner, the trench 150 acousticallyisolates the suspended resonating portion 111 from the support structure110. Additional isolation trenches can be included in the supportstructure.

The resonating portion generally includes a single crystal of apiezoelectric material having any useful crystallographic cut (e.g., anydescribed herein). The dimension(s) of the resonating portion can beselected to optimize the coupling coefficient k_(eff) ², the resonantfrequency f_(s), and/or the quality factor Q. Exemplary dimensionsinclude plate thickness t, width W, and length L (FIG. 1C). Forinstance, to promote particular modes within the resonator (e.g., theSH0 or S0 mode), the thickness-to-wavelength ratio t/λ can be about0.05. The width W can be designed to for a particular wavelength λ(e.g., λ=2 W for a Lamb wave resonator) or for a particular resonantfrequency f_(s)=c/λ W, where c is the relevant sound velocity.

The anchoring region(s) connect the resonating portion to the supportstructure. In addition, the anchoring regions can be designed to suspendand isolate the resonating region. For instance, the microresonator caninclude thin, tapered anchoring regions that suspend the resonatingportion above a trench that is located within the substrate or supportstructure. Exemplary structures for anchoring regions are described inU.S. Pat. No. 8,669,823, which is incorporated herein by reference inits entirety.

In another instance, the resonating region can be a rectangular regionhaving four corners, and an anchoring region can be located on each ofthe corners to connect and suspend the resonating region above a trench(see, e.g., Gong S et al., “Design and analysis of lithium-niobate-basedhigh electromechanical coupling RF-MEMS resonators for widebandfiltering,” IEEE Trans. Microwave Theory Tech. 2013 January;61(1):403-14). Other design and material considerations can beincorporated into the geometry of the anchoring region in order topromote acoustic and/or thermal isolation.

One or more electrodes can be used to drive and/or sense the acousticwaves in the piezoelectric crystal. The electrode(s) can have any usefuldimension and/or orientation. For instance, the electrode can have anyuseful electrode width e (e.g., about 5 μm), gap g between twoelectrodes (e.g., about 5 μm), aperture a (e.g., from about 20 μm toabout 200 μm), and space s (e.g., from about 20 μm to about 200 μm)(FIG. 1C). The electrode can be placed at any useful location of theresonating portion, such as on the top surface and/or at or near theedge of the resonating portion. Additional electrodes can be included(e.g., one or more ground electrodes, opposite polarity electrodes, orfloating electrodes) on one or more surfaces of the resonating portion(e.g., the top and/or bottom surfaces) and/or the anchoring region(s).

The electrode can be electrically connected (e.g., by wiring) to one ormore bond pads (e.g., contact pads and/or ground pads) to provideelectrical input and output connections for the microresonator.Exemplary electrodes include an interdigitated transducer, a gratingelectrode, a thin film electrode, and/or a floating electrode having anyuseful thickness, period, material, or geometric arrangement and formedby any useful process, such as sputtering, vacuum deposition, orion-plating. Exemplary electrodes are described in U.S. Pat. Nos.8,497,747 and 8,522,411, each of which is incorporated herein byreference in its entirety. Non-limiting materials for electrodes includealuminum (Al), titanium (Ti), gold (Au), copper (Cu), tungsten (W),molybdenum (Mo), platinum (Pt), ruthenium (Ru), tantalum (Ta), chromium(Cr), osmium (Os), rhenium (Re), iridium (Ir), as well as alloys, dopedforms, and multilayers thereof (e.g., TiW/AlCu or TiW/Cu layers). Arraysof n electrodes or n pairs of electrodes (e.g., n is 2, 4, 5, 10, 15,20, 24, etc.) can also be incorporated with the microresonator of theinvention. In particular embodiments, the electrode includes a lowerresistive material to improve the quality factor and/or to operate athigher frequencies (e.g., any lower resistive material described herein,such as gold, silver, copper, aluminum, as well as doped forms thereof).

The present method can be used to form any useful resonator. Exemplaryresonators includes a Lamb wave resonator having a plate resonatingportion; a thickness mode acoustic wave resonator having longitudinalelastic waves traveling though the solid material; a contour moderesonator having a suspended resonating portion; as well asthickness-field excitation and lateral field excitation resonatorsthereof. In addition, the resonator can be optimized to promotesymmetric modes, asymmetric modes, and/or shear modes of the acousticwave propagating in the piezoelectric crystal. Spurious modes can bereduced by any useful method, such as by rotating bond/contact pads.

Additional microresonators (e.g., length-extensional resonators, contourmode resonators, thickness mode resonators, ring resonators, and barresonators) and anchoring region designs are described in U.S. Pat. Nos.7,652,547, 8,367,305, and 8,669,823; Wang R et al., “Thin-film lithiumniobate contour-mode resonators,” Proc. 2012 IEEE Intl Ultrason. Symp.(IUS), held on 7-10 Oct. 2012, in Dresden, Germany, pp. 303-6; andKadota M et al., “High-frequency Lamb wave device composed of MEMSstructure using LiNbO₃ thin film and air gap,” IEEE Trans. Ultrason.Ferroelectr. Freq. Control 2010 November; 57(11):2564-71, each of whichis incorporated herein by reference in its entirety.

Single Crystal

The present invention includes microresonators and methods that employ apiezoelectric crystal. In particular embodiments, the crystal is ofcrystallographic class 3 m, such as lithium niobate or lithiumtantalate.

Single crystals are available as plate cuts along a particularcrystallographic axis or axes. Fundamental acoustic waves propagatedifferently through different plate cuts. For instance, in an X-cutlithium niobate plate, shear SH0 waves with a propagation direction thatis 170° from the y-axis have a coupling coefficient K² of about 38. Incontrast, for that same X-cut plate and propagation direction,asymmetric A0 waves have a coupling coefficient K² of about 0.8. Inanother instance, in a Y-cut lithium niobate plate, SH0 waves with apropagation direction that is 0° from the x-axis have a couplingcoefficient K² of about 35. In contrast, for that same Y-cut plate andpropagation direction, A0 waves have a coupling coefficient K² of about4. Accordingly, the particular cut of a single crystal, as well as thepropagation direction (e.g., as controlled by the geometry andarrangement of the driving electrodes), provide microresonators havingdifferent modes and coupling coefficients.

Exemplary cuts include X-cut, Y-cut, and Z-cut plates, as well asrotated cuts. Within these cuts, the acoustic wave propagates at aparticular angle from an axis. For example, FIG. 2 provides a schematicof an X-cut single crystal having wave propagation directions relativeto the z-axis or the y-axis. Based on the location and arrangement ofthe electrodes, the propagation direction of the acoustic wave can becontrolled.

Methods of Fabrication

The present invention includes methods for fabricating a single crystalmicroresonator. In particular, the method relies on ion implantation tofracture a sub-surface portion of the single crystal and on subsequentuse of an etchant to remove that sub-surface portion. In this way, onedimension of the isolated resonating portion (i.e., thickness t) can bedetermined lithographically. This method allows other dimensions of theresonating portion (e.g., W and L dimensions) and electrodes (e.g., a,g, and e dimensions) to be determined lithographically. As describedherein, these dimensions contribute to various physical characteristicsof the microresonator, such as f_(s), k_(eff) ², Q, and FOM. The abilityto control these physical characteristics using lithography providesnumerous benefits, such as the ability to form multiple frequencyfilters on a single die. Additional details are described herein.

FIG. 3A describes an exemplary method for forming a thin plate Lamb wavemicroresonator. The top surface of a substrate 210 (e.g., a singlepiezoelectric crystal, such as any herein) can include any usefulexposed area 275. Optionally, a mask 270 can be patterned on the topsurface to define the exposed area 275. The sub-surface portion of thesubstrate beneath the exposed area will be damaged by ion implantationand ultimately removed. Masked regions will not be exposed to the ionsand, therefore, preserved. The mask can be formed from any usefulmaterial that will not be penetrated by the ions in the ion implantingstep described below. Exemplary mask materials include, e.g., an oxidelayer (e.g., SiO₂), a hard chrome mask, a nickel mask, a gold mask, etc.

Then, the substrate is exposed to an ion source 290 to provide an iondamaged region 276 below the top surface of the crystal. The ion sourcecan be of any useful type (e.g., helium (e.g., He⁺), hydrogen, krypton,xenon, magnesium, fluorine, oxygen (e.g., O³⁺), copper, or gold), energy(e.g., of from about 0.5 MeV to about 100 MeV), and fluence (e.g., offrom about 10¹² to about 10¹⁸ ions/cm² or ions/cm³) provided in one ormore steps to obtain the appropriate penetration depth, which determinesthe thickness t of the resonating portion. In some embodiments, theconditions provide a resonating portion having t of from about 0.2 μm toabout 2 μm (e.g., about 0.5 μm, 1.0 μm, or 1.5 μm).

Next, one or more trenches 251, 252 are provided to define one or moredimensions of the resonator. For the microresonator 200 in FIG. 3A, thedistance between the two trenches determines the width W of theresonating portion, and the length of the trench determines the length Lof the resonating portion. The trenches can be provided using any usefultechnique, such as by patterning the top surface of the substrate with amask (e.g., an oxide hard mask) defining the trench(es), etching (e.g.,dry etching with any useful ion, such as chlorine) the substrate, andthen removing the mask. The distance, geometry, arrangement, number, anddimensions of the trenches can be modified to obtain the desireddimensions for the resonating portion of the microresonator(s). Inaddition, etching conditions can be optimized to ensure that thetrenches provide access to the ion damaged region below the surface ofthe crystal and/or to provide vertical sidewalls for the resonatingportion.

Finally, the ion damaged region 276 is removed using an etchant (e.g., awet etchant), providing a third trench 253 disposed beneath theresonating portion and thereby releasing the resonating portion (here, asuspended plate 211) from the support structure. Any useful etchant canbe employed to remove the damaged region. Exemplary etchants include awet chemical etchant, such as HF and mixtures thereof (e.g., HNO₃ and HFmixtures, optionally including ethanol).

The method can include any number of other useful steps. FIG. 3Bprovides an exemplary method of fabricating a microresonator having apair of electrodes, as described below. The details of FIG. 3C aredescribed in Example 1 herein.

First, as seen in FIG. 3B, a metal layer 320 is deposited on a substrate310 (e.g., a single crystal) by any useful manner (e.g., evaporation,deposition, etc.). Optionally, one or more alignment marks are includedin the metal layer 320. This metal layer 320 can include one or moremetals, metal alloys, or metal layers useful for making an electrode(e.g., any metal described herein).

Next, the top surface of the substrate 310 having the metal layer 320 isoptionally patterned with a mask 370 to define the exposed area 375. Thesubstrate is exposed to an ion source 390 to provide an ion damagedregion 376 below the top surface of the crystal. Next, one or moretrenches 351, 352 are provided to define one or more dimensions of theresonator.

Then, the metal layer is patterned to provide one or more electrodes.For instance, electrodes 341, 342 can then be patterned on thepreviously deposited metal layer 320. This step can include any usefullithographic and microfabrication technique, such as any describedherein. In addition, the electrode pattern can include one or moreuseful dimensions, such as aperture a, gap g, space s, electrode widthe, or any other dimension described herein. The electrode pattern caninclude an array of n electrodes or n pairs of electrodes (e.g., asdescribed herein).

Optionally, a protective layer 375 can be deposited to protect a surfaceof the resonator prior to exposure to an etchant (e.g., a wet etchant).As described herein, particular crystallographic faces can havedifferent etch rates. In particular, the −z face of lithium niobatecrystal etches at a higher rate using an HF etchant, as compared to theother faces. Thus, when the −z face is exposed, then a protective layer(e.g., a gold layer) can be deposited to protect this face.

Finally, the ion damaged region 376 is removed using an etchant (e.g.,HF or any etchant herein), providing a third trench 353 disposed beneaththat resonating portion and thereby releasing the resonating portionfrom the support structure. If the protective layer is used, then thesubsequent step can include a stripping step to strip the protectivelayer.

Additional steps can include annealing (e.g., about <350° C.),polishing, and/or ovenizing (e.g., use of local joule heaters) toresonators, such any described in Kim B et al., “Ovenized and thermallytunable aluminum nitride microresonators,” Proc. 2010 IEEE Ultrason.Symp. (IUS), held on 11-14 Oct. 2010 in San Diego, Calif., pp. 974-8;and U.S. Pat. No. 8,669,823, each of which is incorporated herein byreference in its entirety.

Uses

The microresonators and methods of the present invention can be appliedfor any beneficial use. Exemplary uses include one or more resonators toform a band select filter (e.g., for use in wireless handsets), a filterbank, an oscillator, a sensor, and arrays thereof. For use in an array(e.g., including a plurality of resonators in parallel), each resonatorcan be electrically interconnected (e.g., by way of wires, bias lines,etc.) to provide a composite resonator. The array can also include asubstrate and a plurality of networks formed on the substrate, whereeach network is electrically connected in parallel. The array caninclude at least one input configured to receive an electrical signaland relay this signal to the lattice networks, as well as at least oneoutput to provide a filtered electrical signal. Each network can includeat least one microresonator (e.g., any microresonator described herein,where each microresonator can optionally have a different resonantfrequency or physical dimension). Additional arrays and uses aredescribed in U.S. Pat. Nos. 7,385,334 and 8,497,747; Aigner R, “MEMS inRF filter applications: thin-film bulk acoustic wave technology,”Sensors Update 2003 February; 12(1):175-210; Malocha D C, “SAW/BAWacoustoelectronic technology for filters and communication systems,”Proc. 2010 IEEE 11th Annu. Wireless & Microwave Technol. Conf.(WAMICON), held on 12-13 Apr. 2010 in Melbourne, Fla., pp. 1-7; andYantchev V et al., “Thin film Lamb wave resonators in frequency controland sensing applications: a review,” J. Micromech. Microeng. 2013;23:043001 (14 pp.), each of which is incorporated herein by reference inits entirety.

EXAMPLES Example 1 A High Electromechanical Coupling Coefficient SH0Lamb Wave Lithium Niobate Micromechanical Resonator and a Method forFabrication

Described herein is a high coupling coefficient k_(eff) ²micromechanical resonator based on the propagation of SH0 Lamb waves inthin, suspended plates of single crystal X-cut lithium niobate (LiNbO₃).The thin plates were fabricated without the cumbersome wafer bonding,layer fracturing and chemical mechanical polishing in previouslyreported LiNbO₃ microfabrication approaches. The highest couplingcoefficient was found for resonators with acoustic propagation rotated80° from the z-axis (170° from the y-axis), where a fundamental mode SH0Lamb wave resonator propagating in a 1.2 μm thick plate with a platewidth of 20 μm and a corresponding resonant frequency of 101 MHzachieved a coupling coefficient k_(eff) ²=12.4%, a quality factorQ=1300, and a resonator figure of merit FOM=185. The k_(eff) ² and FOMare among the highest reported for micromechanical resonators.Additional details follow.

Microresonators are miniature acoustic resonators fabricated usingintegrated circuit (IC) microfabrication techniques. Recently,microresonators have become of great interest because the CAD-definableresonant frequency allows many filters spanning from several hundred MHzto several GHz to be realized on a single chip (Piazza G et al.,“Piezoelectric aluminum nitride thin films for microelectromechanicalsystems,” MRS Bull. 2012 November; 37(11):1051-61; and Nguyen C T C,“MEMS technology for timing and frequency control,” IEEE Trans.Ultrason. Ferroelect. Freq. Contr. 2007 February; 54(2):251-70). This isespecially important for next generation cellular handsets, where thegrowing number of frequency bands each typically require a discretefilter die. Microresonator technology provides a potential path forintegrating many band select filters on a single die, thus reducing thesize, cost and complexity of next generation wireless handsets.

Currently, band select filters in cellular handset are realized using acombination of many dies containing bulk (BAW) or surface (SAW) acousticwave resonators (see, e.g., Aigner R, “SAW and BAW technologies for RFfilter applications: A review of the relative strengths and weaknesses,”Proc. 2008 IEEE Ultrason. Symp. held 2-5 Nov. 2008, in Beijing, China,pp. 582-9; Lakin K M, “A review of thin-film resonator technology,” IEEEMicrowave Mag. 2003 December; 4(4):61-7; Ruby R et al., “PCS 1900 MHzduplexer using thin film bulk acoustic resonators (FBARs),” Electron.Lett. 1999 May; 35(10):794-5; Campbell C K, “Applications of surfaceacoustic and shallow bulk acoustic wave devices,” Proc. IEEE 1989October; 77(10):1453-84; and Kadota M, “Development of substratestructures and processes for practical applications of various surfaceacoustic wave devices,” Jpn. J. Appl. Phys. 2005 June; 44(6B):4285-91).

Aluminum nitride (AlN) based BAW resonators and SAW resonators formed inlithium niobate (LiNbO₃) or lithium tantalate (LiTaO₃) have the highcoupling coefficients k_(eff) ² required to achieve the required bandselect filter bandwidths of ˜3% of the filter center frequency and thequality factors required for steep filter roll off all while maintaininglow filter insertion loss. AlN BAW resonators operate based on thethickness of a deposited thin film and require a separate film thicknessfor each filter frequency. This makes integration of multiple frequencyfilters on a single die both challenging and costly. While in theory SAWresonators can support a wide range of frequencies on a single chip, inpractice, the thickness of the metal interdigitated electrodes used totransduce SAW resonator is varied with frequency (Kadota M, Jpn. J.Appl. Phys. 2005 June; 44(6B):4285-91), limiting the range of filterbands that can be covered on a single chip. Furthermore, the low SAWphase velocity limits the application of SAW technology in emerging highfrequency bands above 2.5 GHz (Aigner R, Proc. 2008 IEEE Ultrason. Symp.held 2-5 Nov. 2008, in Beijing, China, pp. 582-9).

Recently, both piezoelectric and electrostatically transducedmicroresonators have been the subject of research (see, e.g., Piazza Get al., MRS Bull. 2012 November; 37(11):1051-6; Kim B et al, “AlNmicroresonator-based filters with multiple bandwidths at lowintermediate frequencies,” J. Microelectromech. Sys. 2013 August;22(4):949-61; Lin C M et al., “Temperature-compensated aluminum nitrideLamb wave resonators,” IEEE Trans. Ultrason. Ferroelect. Freq. Contr.2010 March; 57(3):524-32; Piazza G et al. “Single-chipmultiple-frequency AlN MEMS filters based on contour-mode piezoelectricresonators,” J. Microelectromech. Sys. 2007 April; 16(2):319-28;Abdolvand R et al., “Thin-film piezoelectric-on-silicon resonators forhigh-frequency reference oscillator applications,” IEEE Trans. Ultrason.Ferroelect. Freq. Contr. 2008 December; 55(12):2596-606; Pulskamp J S etal., “Piezoelectric PZT MEMS technologies for small-scale robotics andRF applications,” MRS Bull. 2012 November; 37(11):1062-70; Nguyen C T C,IEEE Trans. Ultrason. Ferroelect. Freq. Contr. 2007 February;54(2):251-70; Casinovi G et al., “Lamb waves and resonant modes inrectangular-bar silicon resonators,” J. Microelectromech. Sys. 2010August; 19(4):827-39; and Weinstein D et al., “Internal dielectrictransduction in bulk-mode resonators,” J. Microelectromech. Sys. 2009December; 18(6):1401-8). By establishing the resonance on a laterallypropagating Lamb wave in a suspended plate with a thickness less than anacoustic wavelength, a wide range of filter frequencies can be achievedon a single wafer by altering the CAD-layout of the devices. The k_(eff)² of electrostatically driven resonators at frequencies relevant tocellular communications are orders of magnitude lower than that requiredfor band select filters (see, e.g., Gong S et al., “Design and analysisof lithium-niobate-based high electromechanical coupling RF-MEMSresonators for wideband filtering,” IEEE Trans. Microwave Theory Tech.2013 January; 61(1):403-14).

Piezoelectric Lamb wave resonators formed in deposited thin films ofaluminum nitride (AlN), zinc oxide (ZnO), and lead zirconate titanate(PZT), while having much higher coupling coefficients thanelectrostatically transduced microresonators, still do not have a highenough coupling coefficient for many of the band select filters inwireless handsets.

In 2001 Kuznetsova et al. (“Investigation of acoustic waves in thinplates of lithium niobate and lithium tantalate,” IEEE Trans. Ultrason.Ferroelect. Freq. Contr. 2001 January; 48(1):322-8) reported thetheoretically large piezoelectric coupling that could be achieved forLamb resonators in thin films of single crystal LiNbO₃. The largestcoupling reported by Kuznetsova et al. was for the SH0 Lamb mode inX-cut LiNbO₃ with a k_(eff) ² of 27.4%. The challenge is in realizingthin, suspended membranes of single crystal piezoelectric materials withmuch higher coupling coefficients than the deposited polycrystallinepiezoelectric thin films such as AlN, ZnO, and PZT.

Recently, symmetric (S0) Lamb wave microresonators have been reported insuspended thin films of single crystal X-cut LiNbO₃ and 136° rotatedY-cut LiNbO₃ (see, e.g., Gong S et al., IEEE Trans. Microwave TheoryTech. 2013 January; 61(1):403-14; and Wang R et al., “High k_(t) ²×Q,multi-frequency lithium niobate resonators,” Proc. 2013 IEEE 26th Int'lConf. Micro Electro Mechanical Systems (MEMS), held 20-24 Jan. 2013 inTaipei, Taiwan, pp. 165-8). Both of the fabrication processes reportedin these publications required bonding of a LiNbO₃ device wafer to ahandle wafer using a glue layer. Thin films of LiNbO₃ directly over theglue/release layer were then realized either by polishing back theLiNbO₃ device wafer to the desired thickness of −1 μm or by fracturingthe LiNbO₃ device wafer just below the wafer surface that was previouslyion implanted to induce a damaged fracture plane (Wang R et al., Proc.2013 IEEE 26th Int'l Conf. Micro Electro Mechanical Systems (MEMS), held20-24 Jan. 2013 in Taipei, Taiwan, pp. 165-8; Gong S et al., IEEE Trans.Microwave Theory Tech. 2013 January; 61(1):403-14; Moulet J S et al.,“High piezoelectric properties in LiNbO₃ transferred layer by the SmartCutTM technology for ultra wide band BAW filter applications,” Proc.2008 IEEE Int'l Electron. Devices Meeting, held 15-17 Dec. 2008 in SanFrancisco, Calif., pp. 1-4; and Aspar B et al., “The generic nature ofthe Smart-Cut® process for thin film transfer,” J. Electron. Mater. 2001July; 30(7):834-40).

In this Example, we provide a Lamb wave LiNbO₃ microresonator fabricatedin a process that does not require costly wafer bonding, polishing orfracturing processes (see, e.g., Yu Y C et al., “Crystal ion-slicinglithium niobate film performed by 250 keV ⁴He ion implantation,” Nucl.Instr. Meth. Phys. Res. B, 2007 March; 256(1):558-60; and Si G et al.,“Suspended slab and photonic crystal waveguides in lithium niobate,” J.Vac. Sci. Technol. B 2010 March; 28(2):316-20). The thin plates werefabricated using ion implantation of He to create an ion damaged layerof LiNbO₃ at a desired depth below the wafer surface. This damaged layerwas selectively wet etched in hydrofluoric (HF) acid based chemistry toform thin, suspended plates of LiNbO₃.

Using this fabrication process, we observed high coupling coefficientSH0 Lamb wave resonators in X-cut LiNbO₃. The highest couplingcoefficient was found for resonators with acoustic propagation rotated80° from the z-axis (see FIG. 2), which is in agreement with the theoryreported in Kuznetsova I E et al., IEEE Trans. Ultrason. Ferroelect.Freq. Contr. 2001 January; 48(1):322-8.

A fundamental mode SH0 Lamb wave resonator propagating in an ˜1.2 μmthick plate with a width of 20 μm and a corresponding resonant frequencyof 101 MHz achieved a k_(eff) ² of 12.4%, a quality factor Q of 1300,and a resonator figure of merit FOM of 185. The k_(eff) ² and FOM areamong the highest reported for micromechanical resonators.

Single Crystal Lithium Niobate Microfabrication Process:

The LiNbO₃ microresonators were fabricated using the process flow shownin FIG. 3C. As shown in step (a), the process began with the evaporationof 100 nm of Cr 420 and the patterning of alignment marks in the Cr onX-cut LiNbO₃ 410.

Next, in step (b), an oxide layer 470 was deposited and patterned todefine an area 475 where an ion implant will penetrate the LiNbO₃,thereby creating an ion damaged LiNbO₃ release layer 480 at the end ofthe ion implant range. Patterning where the ion implant penetrates theLiNbO₃ allowed the lateral extents of the device release to be preciselycontrolled. The sample was then implanted with a He dose of 1×10¹⁶atoms/cm³ at an energy of 0.8 MeV to create an ion damaged release layerof LiNbO₃ 480 approximately 1.8 μm below the wafer surface. The implantswere performed in a 3 MV NEC Pelletron using a current of <7 μA. Tomaintain a low sample temperature, the LiNbO₃ was cooled by liquidnitrogen using a Cu braid during implantation. While the ion implantpasses through the Cr electrodes in the device region, measurementsconfirmed that Cr resistivity was not altered by the ion implant.

Then, in step (c), trenches 451, 452 that define the final resonatordimensions and resonant frequency were etched in the LiNbO₃, therebyexposing the ion damaged LiNbO₃ release layer. The LiNbO₃ was dry etchedat 15° C. using an Ar/BCl₃/Cl₂ gas mixture at 10 mT on a PlasmaThermVersaline 4 in. ICP system using a newly optimized process. This processwas capable of producing sidewalls with >80° sidewall angle and etchdepths >2 μm.

Next, in step (d), the Cr electrodes 421, 422 were patterned.Optionally, as shown in step (e), a 1 μm Au layer was deposited andpatterned to protect the −z face of the LiNbO₃ during the release. Wemeasured the etch rates in the wet hydrofluoric acid release chemistryfor the ion damaged LiNbO₃ and for the different crystal faces of LiNbO₃(Table 1).

TABLE 1 Etch rates of LiNbO₃ using HF LiNbO₃ Etch rate (nm/hr) Iondamaged release layer 9000 +x, −x crystal face 14 +y, −y crystal face204 +z crystal face <14 −z crystal face 2200

Additional studies of etching LiNbO₃ in aqueous HF can be in found in,e.g., Randles A B et al., “Etch rate dependence on crystal orientationof lithium niobate,” IEEE Trans. Ultrason. Ferroelect. Freq. Contr. 2010November; 57(11):2372-80; and Reinisch J et al., “Etching of ionirradiated LiNbO₃ in aqueous hydrofluoric solutions,” J. Electrochem.Soc. 2008 February; 155(4):D298-D301. While the etch rates of the +z, y,and x crystal faces were much lower than the ion damaged LiNbO₃, theetch rate of the −z face was only four times lower than that of the iondamaged LiNbO₃.

Since the highest k_(eff) ² microresonators rotated 80° from the z-axishave a slight −z face component for one of the resonator sidewalls, thisAu layer can be used to protect the −z face and more accurately controlthe final dimensions of the microresonator. In step (f), the device 400was released using a wet hydrofluoric acid chemistry to remove the iondamaged layer of LiNbO₃, thereby forming a trench 453. The devices wereannealed at >300° C. to heal the implant damage in the device layer andflatten the devices. Finally, if utilized, the Au layer used to protectthe −z face of the microresonator sidewall can be stripped in a mixtureof potassium iodide and iodine, KI—I₂. Also shown is the stress field ofthe acoustic standing wave 401.

A scanning electron micrograph (SEM) image of a LiNbO₃ microresonatorwith acoustic propagation rotated 80° from the z-axis is shown in FIG.4A. The designed microresonator dimensions are for a width W=20 μm,length L=140 μm, and an aperture (electrode overlap) a=50 μm. The widthof each Cr electrode e=5 μm, and the gap between electrodes g=5 μm. Theacoustic wave propagated in the width extensional direction, i.e., inthe direction of W.

A zoomed in image of a suspended LiNbO₃ microresonator is shown in FIG.4B, and an optical image of a LiNbO₃ microresonator with acousticpropagation rotated 80° from the z-axis is shown in FIG. 5A. AdditionSEM images of the underside of the LiNbO₃ resonators revealed a smoothsurface and a final plate thickness of approximately 1.2 μm.

FIG. 5B shows an optical image of a LiNbO₃ microresonator with acousticpropagation rotated 70° from the z-axis just prior to release. Thedevice in FIG. 5B has the Au layer from FIG. 3C, step (e), to protectthe −z face of the microresonator during the HF release patterned on thewafer surface. All of the devices shown have identically designeddimensions given in FIG. 4A.

In a microresonator, the acoustic energy is reflected off of an etchedsidewall. In order for the microresonator width W to be nearly identicalthrough the resonator thickness, the etched sidewall angle must be assteep as possible. A significant sidewall angle can both lower theresonator quality factor and introduce spurious responses. Shown in FIG.6 is a cross section image after chlorine dry etching of 1.32 μm ofX-Cut LiNbO₃ rotated 80° from the z-axis using an oxide hard mask. Thesidewall angle is in excess of 84° and is vertical enough to ensure highdevice quality factors.

Lithium Niobate Microresonator Modeling:

The SH0 mode microresonators with acoustic propagation rotated 80° fromthe z-axis formed in X-cut LiNbO₃ were studied using finite elementmodeling (FEM) in ANSYS. For the simulations, the material constantsshown in Table 2 were used (Kovacs G et al., “Improved materialconstants for LiNbO₃ and LiTaO₃ ,” Proc. 1990 IEEE Ultrason. Symp. held4-7 Dec. 1990 in Honolulu, Hi., vol. 1, pp. 435-8) and are slightlydifferent from those reported in Gong S et al., IEEE Trans. MicrowaveTheory Tech. 2013 January; 61(1):403-14.

TABLE 2 Material properties of X-cut LiNbO₃ used for the finite elementmodeling Symbol Value Elastic constants c₁₁ 199.39 (10⁹ N/m²) c₁₂ 54.72c₁₃ 65.13 c₁₄ 7.88 c₃₃ 227.9 c₄₄ 59.65 c₆₆ 72.34 Piezoelectric e₁₅ 3.69constants (C/m²) e₂₂ 2.42 e₃₁ 0.3 e₃₃ 1.77 Dielectric ε11 45.6 constants(F/m) ε33 26.3 Density (kg/m³) ρ 4628

A Young's modulus, density, and Poisson ratio of 279 GPa, 7190 kg/m³,and 0.21 were used to model the Cr electrodes. The devices modeled allhad the dimensions given in FIG. 4A, with a plate width W=20 μm, totallength L=140 μm, and an aperture a=50 μm. All devices modeled had athickness t=1.2 μm, giving a thickness to wavelength ratio (h/λ) of0.03, very close to the optimal value for maximizing k_(eff) ² of 0.05found in Kuznetsova I E et al., IEEE Trans. Ultrason. Ferroelect. Freq.Contr. 2001 January; 48(1):322-8.

The devices were designed to vibrate in the fundamental SH0 mode withthe frequency f_(s) defined by the plate width using Eq. (1):

$\begin{matrix}{{f_{s} = {\frac{C_{SH}}{\lambda} = \frac{C_{SH}}{2W}}},} & (1)\end{matrix}$where C_(SH) is the velocity of the SH0 wave in the thin LiNbO₃ plateand 2 is the acoustic wavelength. The SH0 mode shape for the device inFIG. 4A with 5 μm wide electrodes separated by a gap of 5 μm is shown inFIG. 7. The resonant frequency of 96.12 MHz corresponds to a soundvelocity C_(SH)=3845 m/s using Eq. (1).

The motional impedance of the resonator R_(X) was determined as follows:

$\begin{matrix}{{R_{X} = \frac{1}{2\pi\; f_{s}K^{2}{QC}_{S}}},} & (2)\end{matrix}$

where C_(S) is the capacitance between the electrode fingers in FIG. 3Aand K² is the piezoelectric coupling. C_(S) is directly proportional tothe electrode overlap, or aperture a, and is a complex function of theelectrode width and spacing (Gupta K C et al., “Microstrip Lines andSlotlines: Second Edition,” Artech House Publishers (Norwood, Mass.),1996, p. 133). Since it will be subsequently shown that the electrodespacing also effects the resonator coupling coefficient, a generalapproach is to design the electrode placement for maximum, or lower ifdesired, coupling coefficient and then to adjust the resonator impedanceto the desired value by varying the electrode overlap.

To study the effect of the electrode placement on the resonator couplingcoefficient, harmonic analysis was utilized. A degraded resonatorquality factor of 100 was used in the harmonic simulations to provide afast simulation time. A 5 μm thick layer of vacuum (8=1) was includedabove the resonator, which is important for accurately predicting thecoupling coefficient.

First, the gap between the 100 nm thick Cr electrodes was varied from2.5 μm to 10 μm while holding the electrode width constant at 5 μm, withthe simulated resonator admittances shown in FIG. 8. As the gap betweenthe electrodes increase, the device resonant frequency decreases. Thisis because the electrodes are moving closer to the resonator anti-nodes,where mass loading causes the resonant frequency to be reduced.

The resonator coupling coefficient was then extracted from thesimulations using the IEEE standard (The Institute of Electrical andElectronics Engineers, Inc., “An American National Standard: IEEEStandard on Piezoelectricity,” IEEE (New York, N.Y.), 1988, p. 51):

$\begin{matrix}{{k_{eff}^{2} = \frac{f_{p}^{2} - f_{s}^{2}}{f_{p}^{2}}},} & (3)\end{matrix}$where f_(s) is the frequency of maximum admittance and f_(p) is thefrequency of minimum admittance from FIG. 8. Here, we use k_(eff) ² asthe effective resonator coupling coefficient, which is commonly used infilter design to predict the maximum bandwidth filter that can berealized with a given resonator. The effective resonator couplingcoefficient is related to the piezoelectric coupling K² reported inKuznetsova I E et al., IEEE Trans. Ultrason. Ferroelect. Freq. Contr.2001 January; 48(1):322-8, by following Eq. (4):

$\begin{matrix}{k_{eff}^{2} = {{\frac{K^{2}}{1 + K^{2}}k_{eff}^{2}} = {\frac{K}{1 + K}.}}} & (4)\end{matrix}$

As the electrode gap was increased from 2.5 μm to 15 μm, the couplingcoefficient increased from 11.4% to 24.9%. This suggests that locatingthe electrodes near the resonator edges or anti-nodes results in themaximum coupling coefficient. A similar result was found for overtoneresonators in Wang R et al., Proc. 2013 IEEE 26th Intl Conf. MicroElectro Mechanical Systems (MEMS), held 20-24 Jan. 2013 in Taipei,Taiwan, pp. 165-8. Despite the lower coupling coefficient, theadmittance of the resonators at series resonance was seen to increasefor narrower electrode spacings because of the higher shunt capacitanceand resulting larger transduction force of narrower electrode gaps. Thissuggests that the resonator coupling coefficient and motional impedanceper unit area, which will set the final filter size for a given design,can be traded off to find an optimal implementation.

A similar analysis of k_(eff) ² versus electrode gap was performed forelectrode widths of 2.5 μm and 7.5 μm with the results shown in FIG. 9.The solid lines in FIG. 9 represent the results of harmonic analysissimulations performed with the 5 μm layer of vacuum surrounding themicroresonator device, while the dashed lines are the results when novacuum layer is included in the finite element model. Omitting thevacuum layer over-predicted the coupling coefficient. The maximumk_(eff) ² with the vacuum layer was found to be 25.8%; this is in goodagreement with the theoretical analysis in Kuznetsova I E et al., IEEETrans. Ultrason. Ferroelect. Freq. Contr. 2001 January; 48(1):322-8.

The optimum design to yield the largest coupling coefficient occurredfor an electrode width of 2.5 μm and a gap of 15 μm. This design,however, resulted in electrodes that run directly to the edge of theresonator. For ease of fabrication, we chose an electrode width and gapof 5 μm for our initial experimental studies of SH0 Lamb wave resonatorsin LiNbO₃ plates.

Measured Results and Discussion:

An SH0 mode LiNbO₃ microresonator rotated 80° to the z-axis, such asthat shown in FIG. 4A and FIG. 5A, was measured in vacuum with theresponse shown in black in FIG. 10 (narrow span) and FIG. 11 (widespan). In order to determine the resonator coupling coefficient k_(eff)² and quality factor Q, the measured admittance was fit to the modifiedButterworth-Van Dyke resonator equivalent circuit model shown in FIG. 12using the following Eqs. (5)-(7):

$\begin{matrix}{{C_{X} = \frac{1}{2\pi\; f_{s}{QR}_{X}}},} & (5)\end{matrix}$

$\begin{matrix}{{C_{S} = \frac{\pi^{2}C_{X}}{8k_{t}^{2}}},{and}} & (6)\end{matrix}$

$\begin{matrix}{{L_{X} = \frac{{QR}_{X}}{2\pi\; f_{s}}},} & (7)\end{matrix}$where R_(X) is the measured resonator motional impedance. We note herethe difference between k_(t) ² in Eq. (6) and k_(eff) ² in Eq. (3). Theparasitic capacitance to ground of the bond pads was found to benegligible and was not included in the resonator equivalent circuitmodel.

Excellent agreement between the measured admittance and the electricalequivalent circuit model were found for a motional impedanceR_(X)=1076Ω, a resonator frequency f_(s)=100.965 MHz, a quality factorQ=1300, and a coupling coefficient k_(t) ²=17.5%. These parametersyielded a motional capacitance C_(X)=1.13 fF, a motional inductanceL_(X)=2.21 μH, and an electrical shunt capacitance C_(S)=7.94 fF. Acoupling coefficient k_(eff) ²=12.4% was found for the resonator byusing Eq. (3) and f_(s) and f_(p) from the equivalent circuit model inFIGS. 10-11. The measured quality factor in air was slightly lower,1200, than that measured in vacuum. The measured resonator figure ofmerit FOM was defined by the IEEE standard (The Institute of Electricaland Electronics Engineers, Inc., “An American National Standard: IEEEStandard on Piezoelectricity,” IEEE (New York, N.Y.), 1988, p. 51):

$\begin{matrix}{{FOM} = {\frac{k_{eff}^{2}Q}{1 - k_{eff}^{2}} = {K^{2}{Q.}}}} & (8)\end{matrix}$The measured resonator figure of merit FOM=185 is among the highestrecorded for a Lamb wave resonator.

Numerous spurious resonances with a frequency spacing of approximately4.8 MHz were seen in the wideband microresonator response (FIG. 11).These resonances arose from overtone resonances transduced by the bondpads. Also shown is the measured response between two bond pads rotated170° to the y-axis (gray line, FIG. 11). These bond pads have a membranewithout electrodes placed between the pads to mimic the pad feed throughof the microresonator device as closely as possible. The frequencies ofthe numerous spurious resonances between the bond pads correspondeddirectly to the spurious modes seen for the microresonator device inFIG. 11. We note here that rotating the bond pads to angles with lowcoupling can eliminate these resonant modes by preventing theirtransduction. We also note that no extraction of bond pad or any otherparasitics were performed in this work.

Resonant frequency was measured as a function of temperature. The SH0Lamb wave LiNbO₃ resonator shown in FIG. 4A was measured from 40° C. to100° C. in 20° C. increments (FIG. 13). From these data, a lineartemperature coefficient of frequency of −73 parts-per-million per ° C.(ppm/° C.) was determined for the LiNbO₃ microresonators with thehighest k_(eff) ² with acoustic propagation 80° from the z-axis.

Also shown in FIG. 10 is the predicted admittance from an ANSYS finiteelement harmonic analysis for the SH0 Lamb wave resonator shown in FIG.4A. In the analysis, the resonator quality factor was set to themeasured value of 1300. A 5 μm thick layer of vacuum was included abovethe resonator in the finite element analysis to account for anyreduction in k_(eff) ². The finite element model is in good generalagreement with the measured LiNbO₃ microresonator response. The measuredresonant frequency was slightly higher than that predicted by FEM. Thiscan be attributed to a slightly narrower plate width because the devicereported in FIG. 10 did not include the Au protection layer to preventetching of the −z LiNbO₃ crystal face during release. While the measuredk_(eff) ² of 12.4% is lower than the k_(eff) ² of 17.3% predicted byfitting the equivalent circuit in FIG. 12 to the output of the finiteelement model, it is still among the largest values reported for a Lambwave resonator.

Samples with acoustic propagation 60°, 70°, 80°, and 90° from the z-axiswere fabricated using the fabrication process described in FIG. 3C withthe −z face Au protection layer as shown in FIG. 5B. The measuredadmittances in air for these samples are shown in FIG. 14. Eachmeasurement was fit to the electrical equivalent circuit model in FIG.12, as described above, in order to determine the k_(eff) ² of eachsample.

The highest k_(eff) ² of 10.9% was found for the 80° sample withmeasured k_(eff) ² values of 10.1%, 8.7%, and 6.6% for the 90°, 70°, and60° rotated samples, respectively. The trend of k_(eff) ² versusacoustic propagation direction is in agreement with our own finiteelement modeling and the theoretical analysis in Kuznetsova I E et al.,IEEE Trans. Ultrason. Ferroelect. Freq. Contr. 2001 January;48(1):322-8.

The Q and k_(eff) ² source for the sample with Au −z face protectionwere not as high as the sample reported without it. This, however, wassample specific and is not a result of the −z face protection itself. Weroutinely yielded samples with comparable performance to the devicemeasurement in FIG. 10 using the Au −z face protection. The sample forcomparing k_(eff) ² versus acoustic propagation direction was chosenbecause all four device orientations were relatively spur free, allowingdirect comparison of the k_(eff) ² on a single sample.

The measured resonant frequency of 92.5 MHz for the 80° rotated sampleyielded an acoustic velocity for the SH0 Lamb wave with 100 nm Crelectrodes of 3700 m/s. The measured resonant frequencies 92.48 MHz (80°rotated sample with −z face protection) and 100.965 MHz (without −z faceprotection) predict a final plate width of 18.32 μm for the 80° rotatedsamples released without −z face protection. The characteristics of allthe samples studied in this work are summarized in Table 3.

TABLE 3 Summary of LiNbO₃ microresonator measured performance AcousticAcoustic -z Face propagation propagation protection Coupling Figuredirection direction utilized Center Motional Quality coefficient ofrefereed to refereed to during frequency impedance factor k_(eff) ²Merit the z-axis the y-axis release (MHz) (R_(X)) (Q) (%) (K²Q) 80° 170°No 101.965 1076 1300 12.4 185 80° 170° Yes 92.5 2700 525 10.9 64 70°160° Yes 93.44 3600 510 8.7 44 90° 180° Yes 93.63 5000 450 10.1 46 60°150° Yes 94.11 14500 209 6.6 14

In conclusion, we have presented high coupling coefficient SH0 Lamb wavemicroresonators realized in thin plates of X-cut LiNbO₃. The presentfabrication process eliminated the challenging wafer bonding, polishing,and splitting steps required to realize the S0 Lamb wave LiNbO₃ devicespresented in previous works. The highest coupling coefficient was foundfor resonators with acoustic propagation rotated 80° from the z-axis,which is in agreement with the theory reported in Kuznetsova I E et al.,IEEE Trans. Ultrason. Ferroelect. Freq. Contr. 2001 January;48(1):322-8.

In order to apply this promising technology to RF filteringapplications, the device resonant frequency can be scaled to between 700MHz and 2.4 GHz to cover the most widely utilized commercial RF bands.To reach frequencies as high as 2.4 GHz while maintaining a thickness towavelength ratio <0.25λ required for high coupling coefficient and lowdispersion, the device thickness for an SH0 mode resonator could bescaled to approximately 0.5 μm using the methods described herein.

In addition to scaling the resonant frequency, to realize 50Ω matched,band select filters, the resonator motional impedance must be scaled toon order of 1Ω. Fortunately, when scaling to higher frequencies, themotional impedance per unit area will decrease by 1/f_(s) ² sourceassuming the k_(eff) ² and Q values remain constant. Any additionalscaling of the impedance can be achieved by increasing the electrodeoverlap a, arraying resonators in parallel, or increasing the number ofelectrode fingers. Because both the width of the electrode fingers andthe resonator motional impedance will be decreasing with increasingdevice resonant frequency, replacing the Cr electrodes with a lowerresistivity metal (e.g., any described herein) will be important toprevent degrading the device Q with the series resistance of theelectrodes. With the ability to realize many high k_(eff) ² and high Qmicromechanical resonators operating at different frequencies on asingle chip, single chip filtering solutions covering many cellularbands will become possible, promising to reduce the number of filterchips required in next generation cellular handsets.

Example 2 Lamb Wave S0 and SH0 Micromechanical Resonators Formed in ThinPlates of Lithium Niobate

Microresonator filter arrays have been studied as a smaller, more highlyintegrated replacement for the numerous filters dies that currentlyreside in the RF front-end of a multi-band cellular handset. Inparticular, microresonators realized in thin films of lithium niobate(LiNbO₃) have demonstrated the high piezoelectric coupling needed torealize band select filters with percent bandwidths of 2-5%, whilesimultaneously exhibiting the high quality factors required forduplexers with narrow frequency gaps between the transmit and receivebands (Gong S et al., IEEE Trans. Microwave Theory Tech. 2013 January;61(1):403-14; Wang R et al., Proc. 2013 IEEE 26th Int'l Conf. MicroElectro Mechanical Systems (MEMS), held 20-24 Jan. 2013 in Taipei,Taiwan, pp. 165-8; and Olsson III R H et al., “A high electromechanicalcoupling coefficient SH0 Lamb wave lithium niobate micromechanicalresonator and a method for fabrication,” Sens. Actuat. A 2014 March;209:183-9).

As described herein, we studied and compared the properties of Lamb waveresonators vibrating in the fundamental symmetric (S0) and shear (SH0)modes. These modes were chosen because they are predicted to have bothlow dispersion and high coupling coefficient over a wide range ofthickness-to-wavelength ratios (h/λ). Both of these properties areimportant for realizing multi-frequency band select filters in a singleLiNbO₃ layer.

Fundamental mode bar resonators were realized on a single die for directcomparison. These resonators included a plate width W=20 μm, a platethickness t=1.5 μm, various apertures a (i.e., a=50 μm, 90 μm, and 130μm) and various acoustic wave propagations (i.e., rotated 30° (S0) and170° (SH0) to the +y-axis to maximize piezoelectric coupling). The ha of0.04 is very close to the optimum value to maximize piezoelectriccoupling of 0.05 found for both the S0 and SH0 modes in Kuznetsova I Eet al., IEEE Trans. Ultrason. Ferroelect. Freq. Contr. 2001 January;48(1):322-8.

We found that while the S0 Lamb wave had a 1.6 times higher soundvelocity than the SH0 mode (compare ˜6400 m/s for S0 and 3900 m/s forSH0), the SH0 mode was predicted to and consistently exhibited a 1.6-1.8times higher effective piezoelectric coupling coefficient k_(eff) ². TheSH0 mode also exhibited higher quality factor Q, higher figure-of-meritFOM, and fewer spurious responses.

Finally, a fundamental SH0 mode Lamb wave resonator realized in a 4.4 μmwide plate was demonstrated with an operating frequency of 350 MHz, ak_(eff) ²=14.5%, Q=2150 in air, and a FOM=365, among the highestreported for Lamb wave resonators. Details follow.

Device Fabrication:

The resonators were fabricated using a process flow similar to thatshown in FIG. 3, where selective ion irradiation of a LiNbO₃ wafercreated a damaged release layer that selectively etches in HF chemistry,allowing suspended membranes to be formed. The advantages of thisfabrication process include the following: 1) the ability tolithographically define the undercut of the device, 2) no wafer bonding,polishing or fracturing, and 3) the ability to realize custom andpotentially multiple LiNbO₃ thicknesses on a single substrate.

The process began with an X-cut LiNbO₃ wafer upon which a 100 nm layerof Cr was deposited and patterned to form alignment marks. Next, a SiO₂layer was deposited and patterned to determine where the release layerwill be formed via ion irradiation. The sample was then implanted with aHe dose of 1×10¹⁶ atoms/cm³ at an energy of 0.9 MeV to create an iondamaged release layer of LiNbO₃ approximately 2 μm below the wafersurface. This energy was found to give a final plate thickness afterrelease of approximately 1.5 μm. The implants were performed in a 3 MVNEC Pelletron using a current of <7 μA. A low sample temperature wasmaintained via liquid nitrogen cooling using a Cu braid duringimplantation. The ion implant was intentionally performed through the Crelectrode layer to promote adhesion. After ion implantation, the SiO₂implant masking layer was stripped, and the LiNbO₃ was patterned usingan oxide hard mask and Cl dry etching to define the final platedimensions.

Next, the Cr electrodes were patterned, and a layer of Au was depositedand patterned via lift off to protect the −z face of the LiNbO₃ device,which can have a significant etch rate in HF during the release. Duringour research, we found that the structural rigidity of this Auprotection layer also led to significantly improved device yield throughthe release process. Finally, the devices were released in a HF basedchemistry and the Au protect layer is stripped in KI—I₂.

Device Structure and Dimensions:

The resonator included a 1.5 μm thick suspended membrane of LiNbO₃ witha width W=20 μm. The device was designed to resonate in the fundamentalS0 or SH0 mode, see FIG. 15, with a resonant frequency f_(s)=c/2 W,where c is the sound velocity of the Lamb wave. Thethickness-to-wavelength ratio t/λ=0.04 was close to the optimum value tomaximize k_(eff) ² of 0.05 (Kuznetsova I E et al., IEEE Trans. Ultrason.Ferroelect. Freq. Contr. 2001 January; 48(1):322-8). The space betweenthe electrodes and device substrate anchor s was 45 μm for all devices.While it was found in Wang R et al., Proc. 2013 IEEE 26th Int'l Conf.Micro Electro Mechanical Systems (MEMS), held 20-24 Jan. 2013 in Taipei,Taiwan, pp. 165-8 and Olsson III R H et al., Sens. Actuat. A 2014 March;209:183-90 that to maximize k_(eff) ² the electrodes should be placed atthe edges of the resonator, 5 μm wide electrodes with a gap g=5 μm weredesigned slightly offset from the resonator edge for ease offabrication. Electrode apertures a, which define the resonator staticcapacitance, of 50 μm, 90 μm, and 130 μm were modeled, fabricated, andcharacterized.

Finite Element Modeling Results:

The six devices with three different apertures and two differentrotations to the +y-axis described above were studied using finiteelement modeling (FEM) and experimentally. The FEM was a full 3Drepresentation of the device including the anchors and electrodes.

FIG. 15 shows the displacement and strain profiles from ANSYS FEM forthe S0 Lamb wave resonator rotated 30° to the +y-axis and for the SH0Lamb wave resonator rotated 170° (SH0) to the +y-axis, both withaperture a=50 μm. The maximum strain appeared between the Cr electrodesand was thus efficiently transduced by the electric field applied acrossthese same electrodes. The strain between the electrodes at the edges ofthe aperture for the S0 Lamb wave resonator was seen to dramaticallydecrease, indicating that the S0 Lamb wave k_(eff) ² will besignificantly impacted by the device aperture. The k_(eff) ² for eachdevice was studied using harmonic analysis in ANSYS.

The piezoelectric coupling K² was calculated from the simulatedresonator admittance using Eq. 9:

$\begin{matrix}{{K^{2} = {{\frac{1}{2\pi\; f_{s}{QR}_{x}C_{S}}K} = {{\frac{1}{2\pi\; f_{s}{QR}_{x}C_{S}}K^{2}} = \frac{1}{2\pi\; f_{s}{QR}_{x}C_{S}}}}},} & (9)\end{matrix}$where f_(s) is the frequency of minimum resonator admittance, R_(X) isthe resonator motional impedance, C_(s) is the resonator staticcapacitance, and Q is the resonator quality factor, which is an input tothe FEM. The effective piezoelectric coupling k_(eff) ² was thencalculated using Eq. 4:

$\begin{matrix}{k_{eff}^{2} = {{\frac{K^{2}}{1 + K^{2}}k_{eff}^{2}} = {\frac{K}{1 + K}.}}} & (4)\end{matrix}$In Example 1, k_(eff) ² was calculated from the simulated resonatoradmittance using Eq. 3:

$\begin{matrix}{{k_{eff}^{2} = \frac{f_{p}^{2} - f_{s}^{2}}{f_{p}^{2}}},} & (3)\end{matrix}$where f_(p) is the frequency of maximum resonator admittance.

The results of Eqs. 3 and 4 were equal when no spurious resonances areseen near f_(s) or f_(p). As discussed in this Example, however, wefound in both FEM and in experiments that spurious modes caused asignificant increase in f_(p) and overestimation of k_(eff) ² if Eq. 3is used. We note here that the piezoelectric coupling coefficient k_(t)² reported in Gong S et al., IEEE Trans. Microwave Theory Tech. 2013January; 61(1):403-14 is equal to Eq. 10:

$\begin{matrix}{k_{t}^{2} = {{\frac{\pi^{2}}{8}\frac{C_{X}}{C_{S}}} = {{\frac{\pi^{2}}{8}K^{2}} = {\frac{\pi^{2}}{8}{\frac{k_{eff}^{2}}{1 - k_{eff}^{2}}.}}}}} & (10)\end{matrix}$

The simulated k_(eff) ² versus aperture is shown in FIG. 16 andsummarized in Table 4 for both the S0 and SH0 Lamb wave resonators. TheS0 Lamb wave resonator was predicted to have a significantly lowerk_(eff) ² that is much more sensitive to the aperture than the SH0 Lambwave resonator.

Experimental Results:

The admittance of the six different resonators realized on the same diewas measured in air using a network analyzer. The responses for the S0and SH0 mode resonators with an aperture of 90 μm are shown in FIG. 18(S0) and in FIG. 19 (SH0). Also shown in FIGS. 18-19 are the results ofthe FEM for each device simulated with the measured quality factors.Each resonator measurement was fit to the modified Butterworth Van Dyke(MBVD) equivalent circuit model shown in FIG. 17 using Eqs. 11-13:

$\begin{matrix}{{R_{X} = \frac{1}{2\pi\; f_{s}C_{S}K^{2}Q_{A}}},} & (11)\end{matrix}$

$\begin{matrix}{{C_{X} = \frac{C_{S}}{K^{2}}},{and}} & (12)\end{matrix}$

$\begin{matrix}{{L_{X} = {\frac{R_{x}Q_{A}}{2\pi\; f_{s}} = \frac{( {R_{x} + R_{S}} )Q_{Total}}{2\pi\; f_{s}}}},} & (13)\end{matrix}$

where C_(X) and L_(X) are the motional capacitance and inductance, R_(S)is the series electrical resistance, Q_(A) is the acoustic qualityfactor, and Q_(Total) is the measured 3 dB bandwidth of the acousticadmittance divided by f_(s), which includes the losses from both R_(X)and R_(S). R_(S) was measured directly on a separate test structureallowing it to be extracted from R_(X). The simulated response of theMBVD electrical equivalent circuit model for the S0 and SH0 modemicromechanical resonators with an aperture of 90 μm are shown in FIGS.18-19 along with the equivalent circuit parameters.

The simulated MBVD results and experimental measurements were in goodagreement far from resonance and close to the series resonance. Theexperimental k_(eff) ² for each resonator was calculated from the MBVDequivalent circuit model using either Eq. 3 or 4, which yieldedidentical results since no spurious responses are modeled using thecircuit in FIG. 17. From FIG. 18, it is apparent that using the measuredfrequency of maximum admittance in Eq. 3 dramatically overestimatedk_(eff) ² due to the spurious modes between f_(s) and f_(p).

The measured f_(s), R_(S), R_(X), C_(S), k_(eff) ², k_(t) ², Q_(Total),Q_(A), and acoustic figure-of-merit are summarized for all theresonators in Table 4, while the measured k_(eff) ² versus aperture isshown in FIG. 16. The SH0 resonators were found to have significantlyhigher effective piezoelectric coupling coefficient and figure of meritvalues.

TABLE 4 Experimental and FEM results for the S0 and SH0 Lamb wavemicroresonators FEM Measured Measured Orientation a Freq. RS RX CSk_(eff) ² k_(eff) ² k_(t) ² Mode to +y-axis (μm) (MHz) (Ω) (Ω) (fF) (%)(%) (%) Q_(Total) Q_(A) FOM* S0  30° 50 158.9 194 3516 9 12.4 5.6 7.4500 528 31 S0  30° 90 160.6 233 1386 12 16.2 7.8 10.5 600 701 59 S0  30°130 161.5 272 944 18 15.0 9.1 12.3 450 580 58 SH0 170° 50 98.4 194 9228.5 24.0 13.4 19.1 1100 1331 206 SH0 170° 90 96.9 233 702 12 25.5 16.324.1 750 999 195 SH0 170° 130 94.8 272 783 17 25.8 16.3 24.1 480 647 126*k_(eff) ²Q_(A)/(1 − k_(eff) ²)

Scaling to Higher Frequencies:

From Table 4, the resistance of the Cr electrodes was seen tosignificantly degrade the total resonator quality factor. Higherfrequency operation can only result in further degradation in Q_(Total)as the electrode cross sectional area will decrease, causing R_(S) toincrease and R_(X) to decrease (see Eq. 11). For this reason, whenscaling to higher operating frequencies, the Cr electrodes were replacedwith 100 nm of Au (lower resistivity compared to Cr), and the Au −z faceprotection mask step was omitted from the fabrication process.

The measured admittance of a SH0 Lamb wave resonator rotated 170° to the+y-axis with a plate width W=4.4 μm, an aperture a=60 μm, an electrodewidth e=1 μm, and a gap between the electrodes g=2 μm, is shown in FIG.20A-20B. Also shown in both figures is the simulated response of theMBVD equivalent circuit. The narrowing of the plate width W to 4.4 μmincreased the series resonant frequency to 350 MHz. The inclusion of theAu electrodes has reduced the series electrical resistance to the pointwhere it need not be modeled, R_(S)=0Ω and Q_(A)=Q_(Total). The increasein the effective electrode aperture to nearly seven wavelengths hasenabled a corresponding increase in the device quality factor to 2150.The ultra-high frequency (UHF) band resonator maintained a higheffective coupling coefficient k_(eff) ²=14.5% and a figure of meritFOM=365. The performance of the UHF, fundamental SH0 mode LiNbO₃micromechanical resonator is summarized in Table 5.

TABLE 5 Experimental results for a SH0 Lamb wave resonator operating inUHF band Plate Measured Measured TFOM Orientation Width Aperture Freq.R_(X) C_(S) k_(eff) ² k_(t) ² k_(eff) ²Q/ Mode to +y-axis W (μm) a (μm)(MHz) (Ω) (fF) (%) (%) Q (1 − k_(eff) ²) SH0 170° 4.4 64 350 78 16 14.521 2150 365

Conclusion:

We have studied, theoretically and experimentally, fundamental mode S0and SH0 Lamb wave resonators realized in thin plates of LiNbO₃. Thedevices were fabricated using a newly developed process that allows theformation of a damaged LiNbO₃ sacrificial layer using helium ionimplantation. This damaged LiNbO₃ was subsequently etched in a HF acidbased wet release. The plate width, which determines the resonantfrequency, was 20 μm wide for both the S0 and SH0 mode resonators, andthe thickness-to-wavelength ratio for both types of resonators was 0.04,near the optimum value to maximize piezoelectric coupling found inKuznetsova I E et al., IEEE Trans. Ultrason. Ferroelect. Freq. Contr.2001 January; 48(1):322-8. The acoustic wave propagation was rotated 30°(S0) and 170° (SH0) to the +y-axis for the resonators, also an optimumcondition to maximize piezoelectric coupling.

We found that the SH0 mode microresonators consistently exhibited highereffective piezoelectric coupling, quality factor, and figure of meritwhen compared to identically designed S0 mode structures. The propertiesof the SH0 mode resonators were less sensitive to the device apertureand the SH0 mode resonators exhibited fewer spurious responses. Whilethe initial study was performed on LiNbO₃ microresonators operating at97 MHz (SH0) and 160 MHz (S0), a SH0 mode microresonator in a 4.4 μmwide LiNbO₃ plate were demonstrated at 350 MHz. The k_(eff) ²=14.5%,Q=2150, and FOM=365 for the 350 MHz microresonator are among the highestdemonstrated for this new class of resonant microdevices.

Other Embodiments

All publications, patents, and patent applications mentioned in thisspecification are incorporated herein by reference to the same extent asif each independent publication or patent application was specificallyand individually indicated to be incorporated by reference.

While the invention has been described in connection with specificembodiments thereof, it will be understood that it is capable of furthermodifications and this application is intended to cover any variations,uses, or adaptations of the invention following, in general, theprinciples of the invention and including such departures from thepresent disclosure that come within known or customary practice withinthe art to which the invention pertains and may be applied to theessential features hereinbefore set forth, and follows in the scope ofthe claims.

Other embodiments are within the claims.

The invention claimed is:
 1. A method for fabricating a micromechanicalresonator, the method comprising: (i) providing a single crystalcomprising lithium niobate or lithium tantalate; (ii) treating anexposed area of the single crystal with ions, thereby creating an iondamaged region below a top surface of the crystal; (iii) providing atleast one trench that defines a first dimension of the resonator; (iv)removing the ion damaged layer with an etchant, thereby releasing atleast a resonating portion of the resonator from the crystal; and (v)annealing the resonator.
 2. The method of claim 1, further comprising,before step (ii), patterning a top surface of the crystal with a mask,thereby defining the exposed area.
 3. The method of claim 2, wherein themask comprises a plurality of exposed areas.
 4. The method of claim 3,wherein each exposed area defines a resonating portion of a resonator,thereby providing a plurality of micromechanical resonators on a singledie.
 5. The method of claim 4, wherein two or more of the plurality ofmicromechanical resonators are the same or different.
 6. The method ofclaim 1, further comprising, before step (iv), depositing a protectivelayer on the first dimension of the resonator.
 7. The method of claim 6,further comprising, after step (iv), stripping the protective layer. 8.The method of claim 1, further comprising, after step (i), depositing ametal layer on a top surface of the crystal, wherein a mask is thenpatterned on top of the metal layer.
 9. The method of claim 8, furthercomprising, after step (ii), patterning the metal layer with one or moreelectrodes.
 10. The method of claim 1, wherein the crystal is an X-cutlithium niobate crystal, a Y-cut lithium niobate crystal, a Z-cutlithium niobate crystal, a rotated cut lithium niobate crystal, an X-cutlithium tantalate crystal, a Y-cut lithium tantalate crystal, a Z-cutlithium tantalate crystal, or a rotated-cut lithium tantalate crystal.11. The method of claim 10, wherein the resonator is a shear mode Lambwave resonator.
 12. The method of claim 10, wherein the resonator is asymmetric mode Lamb wave resonator.
 13. The method of claim 1, whereinthe ion is helium or hydrogen.
 14. A method for fabricating amicromechanical resonator, the method comprising: providing a singlecrystal comprising lithium niobate or lithium tantalate; treating anexposed area of the single crystal with ions, thereby creating an iondamaged region below a top surface of the crystal; providing at leastone trench that defines a first dimension of the resonator; depositing aprotective layer on the first dimension of the resonator; and removingthe ion damaged layer with an etchant, thereby releasing at least aresonating portion of the resonator from the crystal.
 15. The method ofclaim 14, further comprising patterning a top surface of the crystalwith a mask.
 16. A method for fabricating a micromechanical resonator,the method comprising: providing a single crystal; treating an exposedarea of the single crystal with ions, thereby creating an ion damagedregion below a top surface of the crystal; depositing a metal layer onthe top surface of the crystal; providing at least one trench thatdefines a first dimension of the resonator; and removing the ion damagedlayer with an etchant.
 17. The method of claim 16, further comprisingpatterning the metal layer with one or more electrodes.